From Surf Wiki (app.surf) — the open knowledge base

Special classes of semigroups

Families of certain algebraic structures

In mathematics, a semigroup is a nonempty set together with an associative binary operation. A special class of semigroups is a class of semigroups satisfying additional properties or conditions. Thus the class of commutative semigroups consists of all those semigroups in which the binary operation satisfies the commutativity property that ab = ba for all elements a and b in the semigroup. The class of finite semigroups consists of those semigroups for which the underlying set has finite cardinality. Members of the class of Brandt semigroups are required to satisfy not just one condition but a set of additional properties. A large collection of special classes of semigroups have been defined though not all of them have been studied equally intensively.

In the algebraic theory of semigroups, in constructing special classes, attention is focused only on those properties, restrictions and conditions which can be expressed in terms of the binary operations in the semigroups and occasionally on the cardinality and similar properties of subsets of the underlying set. The underlying sets are not assumed to carry any other mathematical structures like order or topology.

As in any algebraic theory, one of the main problems of the theory of semigroups is the classification of all semigroups and a complete description of their structure. In the case of semigroups, since the binary operation is required to satisfy only the associativity property the problem of classification is considered extremely difficult. Descriptions of structures have been obtained for certain special classes of semigroups. For example, the structure of the sets of idempotents of regular semigroups is completely known. Structure descriptions are presented in terms of better known types of semigroups. The best known type of semigroup is the group.

A (necessarily incomplete) list of various special classes of semigroups is presented below. To the extent possible the defining properties are formulated in terms of the binary operations in the semigroups. The references point to the locations from where the defining properties are sourced.

Notations

In describing the defining properties of the various special classes of semigroups, the following notational conventions are adopted.

Notation	Meaning
S	Arbitrary semigroup
E	Set of idempotents in S
G	Group of units in S
I	Minimal ideal of S
V	Regular elements of S
X	Arbitrary set
a, b, c	Arbitrary elements of S
x, y, z	Specific elements of S
e, f, g	Arbitrary elements of E
h	Specific element of E
l, m, n	Arbitrary positive integers
j, k	Specific positive integers
v, w	Arbitrary elements of V
0	Zero element of S
1	Identity element of S
S1	S if 1 ∈ S; S ∪ { 1 } if 1 ∉ S
a ≤L b
a ≤R b
a ≤H b
a ≤J b	S1a ⊆ S1b
aS1 ⊆ bS1
S1a ⊆ S1b and aS1 ⊆ bS1
S1aS1 ⊆ S1bS1
L, R, H, D, J	Green's relations
La, Ra, Ha, Da, Ja	Green classes containing a
x^\omega	The only power of x which is idempotent. This element exists, assuming the semigroup is (locally) finite. See variety of finite semigroups for more information about this notation.
	X	The cardinality of X, assuming X is finite.

For example, the definition xab = xba should be read as:

There exists x an element of the semigroup such that, for each a and b in the semigroup, xab and xba are equal.

List of special classes of semigroups

The third column states whether this set of semigroups forms a variety. And whether the set of finite semigroups of this special class forms a variety of finite semigroups. Note that if this set is a variety, its set of finite elements is automatically a variety of finite semigroups.

Terminology	Defining property	Variety of finite semigroup	Reference(s)
Finite semigroup
Empty semigroup		No
Trivial semigroup
Monoid		No	Gril p. 3
Band
(Idempotent semigroup)			C&P p. 4
Rectangular band			Fennemore
Normal band			Fennemore
Semilattice	A commutative band, that is:
Commutative semigroup			C&P p. 3
Archimedean commutative semigroup			C&P p. 131
Nowhere commutative semigroup			C&P p. 26
Left weakly commutative			Nagy p. 59
Right weakly commutative			Nagy p. 59
Weakly commutative	Left and right weakly commutative. That is:		Nagy p. 59
Conditionally commutative semigroup			Nagy p. 77
R-commutative semigroup			Nagy p. 69–71
RC-commutative semigroup			Nagy p. 93–107
L-commutative semigroup			Nagy p. 69–71
LC-commutative semigroup			Nagy p. 93–107
H-commutative semigroup			Nagy p. 69–71
Quasi-commutative semigroup			Nagy p. 109
Right commutative semigroup			Nagy p. 137
Left commutative semigroup			Nagy p. 137
Externally commutative semigroup			Nagy p. 175
Medial semigroup			Nagy p. 119
E-k semigroup (k fixed)			Nagy p. 183
Exponential semigroup			Nagy p. 183
WE-k semigroup (k fixed)			Nagy p. 199
Weakly exponential semigroup			Nagy p. 215
Right cancellative semigroup			C&P p. 3
Left cancellative semigroup			C&P p. 3
Cancellative semigroup	Left and right cancellative semigroup, that is		C&P p. 3
E-inversive semigroup (E-dense semigroup)			C&P p. 98
Regular semigroup			C&P p. 26
Regular band			Fennemore
Intra-regular semigroup			C&P p. 121
Left regular semigroup			C&P p. 121
Left-regular band			Fennemore
Right regular semigroup			C&P p. 121
Right-regular band			Fennemore
Completely regular semigroup			Gril p. 75
(inverse) Clifford semigroup			Petrich p. 65
k-regular semigroup (k fixed)			Hari
Eventually regular semigroup
(π-regular semigroup,
Quasi regular semigroup)			Edwa
Shum
Higg p. 49
Quasi-periodic semigroup, epigroup, group-bound semigroup, completely (or strongly) π-regular semigroup, and many other; see Kela for a list)			Kela
Gril p. 110
Higg p. 4
Primitive semigroup			C&P p. 26
Unit regular semigroup			Tvm
Strongly unit regular semigroup			Tvm
Orthodox semigroup			Gril p. 57
Howi p. 226
Inverse semigroup			C&P p. 28
Left inverse semigroup
(R-unipotent)			Gril p. 382
Right inverse semigroup
(L-unipotent)			Gril p. 382
Locally inverse semigroup
(Pseudoinverse semigroup)			Gril p. 352
M-inversive semigroup			C&P p. 98
Abundant semigroup			Chen
Rpp-semigroup
(Right principal projective semigroup)			Shum
Lpp-semigroup
(Left principal projective semigroup)			Shum
Null semigroup
(Zero semigroup)			C&P p. 4
Left zero semigroup			C&P p. 4
Left zero band	A left zero semigroup which is a band. That is:
Left group			C&P p. 37, 38
Right zero semigroup			C&P p. 4
Right zero band	A right zero semigroup which is a band. That is:		Fennemore
Right group			C&P p. 37, 38
Right abelian group			Nagy p. 87
Unipotent semigroup			C&P p. 21
Left reductive semigroup			C&P p. 9
Right reductive semigroup			C&P p. 4
Reductive semigroup			C&P p. 4
Separative semigroup			C&P p. 130–131
Reversible semigroup			C&P p. 34
Right reversible semigroup			C&P p. 34
Left reversible semigroup			C&P p. 34
Aperiodic semigroup
ω-semigroup			Gril p. 233–238
Left Clifford semigroup
(LC-semigroup)			Shum
Right Clifford semigroup
(RC-semigroup)			Shum
Orthogroup			Shum
Complete commutative semigroup			Gril p. 110
Nilsemigroup (Nilpotent semigroup)
Elementary semigroup			Gril p. 111
E-unitary semigroup			Gril p. 245
Finitely presented semigroup			Gril p. 134
Fundamental semigroup			Gril p. 88
Idempotent generated semigroup			Gril p. 328
Locally finite semigroup			Gril p. 161
N-semigroup			Gril p. 100
L-unipotent semigroup
(Right inverse semigroup)			Gril p. 362
R-unipotent semigroup
(Left inverse semigroup)			Gril p. 362
Left simple semigroup			Gril p. 57
Right simple semigroup			Gril p. 57
Subelementary semigroup			Gril p. 134
Symmetric semigroup
(Full transformation semigroup)			C&P p. 2
Weakly reductive semigroup			C&P p. 11
Right unambiguous semigroup			Gril p. 170
Left unambiguous semigroup			Gril p. 170
Unambiguous semigroup			Gril p. 170
Left 0-unambiguous			Gril p. 178
Right 0-unambiguous			Gril p. 178
0-unambiguous semigroup			Gril p. 178
Left Putcha semigroup			Nagy p. 35
Right Putcha semigroup			Nagy p. 35
Putcha semigroup			Nagy p. 35
Bisimple semigroup
(D-simple semigroup)			C&P p. 49
0-bisimple semigroup			C&P p. 76
Completely simple semigroup			C&P p. 76
Completely 0-simple semigroup			C&P p. 76
D-simple semigroup
(Bisimple semigroup)			C&P p. 49
Semisimple semigroup			C&P p. 71–75
\mathbf{CS}: Simple semigroup
0-simple semigroup			C&P p. 67
Left 0-simple semigroup			C&P p. 67
Right 0-simple semigroup			C&P p. 67
Cyclic semigroup
(Monogenic semigroup)			C&P p. 19
Periodic semigroup			C&P p. 20
Bicyclic semigroup			C&P p. 43–46
Full transformation semigroup TX
(Symmetric semigroup)			C&P p. 2
Rectangular band			Fennemore
Rectangular semigroup			C&P p. 97
Symmetric inverse semigroup IX			C&P p. 29
Brandt semigroup			C&P p. 101
Free semigroup FX			Gril p. 18
Rees matrix semigroup			C&P p.88
Semigroup of linear transformations			C&P p.57
Semigroup of binary relations BX			C&P p.13
Numerical semigroup			Delg
Semigroup with involution
(*-semigroup)			Howi
Baer–Levi semigroup			C&P II Ch.8
U-semigroup			Howi p.102
I-semigroup			Howi p.102
Semiband			Howi p.230
Group
Topological semigroup			Pin p. 130
Syntactic semigroup			Pin p. 14
\mathbf R: the R-trivial monoids			Pin p. 158
\mathbf L: the L-trivial monoids			Pin p. 158
\mathbf J: the J-trivial monoids			Pin p. 158
\mathbf{R_1}: idempotent and R-trivial monoids			Pin p. 158
\mathbf {L_1}: idempotent and L-trivial monoids			Pin p. 158
\mathbb D\mathbf{S}: Semigroup whose regular D are semigroup			Pin pp. 154, 155, 158
\mathbb D\mathbf{A}: Semigroup whose regular D are aperiodic semigroup			Pin p. 156, 158
\ell\mathbf{1}/\mathbf K: Lefty trivial semigroup			Pin pp. 149, 158
\mathbf{r1}/\mathbf D: Right trivial semigroup			Pin pp. 149, 158
\mathbb L\mathbf{1}: Locally trivial semigroup			Pin pp. 150, 158
\mathbb L\mathbf{G}: Locally groups			Pin pp. 151, 158

Terminology	Defining property	Variety	Reference(s)
Ordered semigroup			Pin p. 14
\mathbf{N}^+			Pin pp. 157, 158
\mathbf{N}^-			Pin pp. 157, 158
\mathbf{J}_1^+			Pin pp. 157, 158
\mathbf{J}_1^-			Pin pp. 157, 158
\mathbb L\mathbf{J}_1^+ locally positive J-trivial semigroup			Pin pp. 157, 158

References

Info: Wikipedia Source

This article was imported from Wikipedia and is available under the Creative Commons Attribution-ShareAlike 4.0 License. Content has been adapted to SurfDoc format. Original contributors can be found on the article history page.

algebraic-structures semigroup-theory

Want to explore this topic further?

Ask Mako anything about Special classes of semigroups — get instant answers, deeper analysis, and related topics.

Research with Mako

Free with your Surf account

Content sourced from Wikipedia, available under CC BY-SA 4.0.

This content may have been generated or modified by AI. CloudSurf Software LLC is not responsible for the accuracy, completeness, or reliability of AI-generated content. Always verify important information from primary sources.

Report